Infinitesimal Morita homomorphisms and the tree-level of the LMO invariant

Gwenael Massuyeau (1)

1 IRMA - Institut de Recherche Mathématique Avancée

Let S be a compact connected oriented surface with one boundary component, and let P be the fundamental group of S. The Johnson filtration is a decreasing sequence of subgroups of the Torelli group of S, whose k-th term consists of the self-homeomorphisms of S that act trivially at the level of the k-th nilpotent quotient of P. Morita defined a homomorphism from the k-th term of the Johnson filtration to the third homology group of the k-th nilpotent quotient of P. In this paper, we replace groups by their Malcev Lie algebras and we study the "infinitesimal" version of the k-th Morita homomorphism, which corresponds to the original version by a canonical isomorphism. We give a diagrammatic description of the k-th infinitesimal Morita homomorphism and, given an expansion of the free group P which is "symplectic" in some sense, we show how to compute it from Kawazumi's total Johnson map. We also consider the diagrammatic representation of the Torelli group that we derived from the Le-Murakami-Ohtsuki invariant of 3-manifolds in a previous joint work with Cheptea and Habiro, and which we call the "LMO homomorphism." We show that the Le-Murakami-Ohtsuki invariant induces a symplectic expansion of P for which the total Johnson map tantamounts to the tree-reduction of the LMO homomorphism. We deduce that the k-th infinitesimal Morita homomorphism coincides with the degree [k,2k[ part of the tree-reduction of the LMO homomorphism. Our results also apply to the monoid of homology cylinders over S.

Domaines: Topologie géométrique [math.GT]; Topologie algébrique [math.AT]; Algèbres quantiques [math.QA]

Gwenael Massuyeau. Infinitesimal Morita homomorphisms and the tree-level of the LMO invariant. Bulletin de la société mathématique de France, 2012, 140 (1), pp.101-161. ⟨10.24033/bsmf.2625⟩. ⟨hal-00325341⟩